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On the umbilic set of immersed surfaces in three-dimensional space forms
Bulletin des Sciences Mathématiques ( IF 1.3 ) Pub Date : 2020-10-24 , DOI: 10.1016/j.bulsci.2020.102917
Giovanni Catino , Alberto Roncoroni , Luigi Vezzoni

We prove that, under some integral inequality regarding the mean curvature and the traceless second fundamental form, the set of umbilical points of an immersed surface in a 3-dimensional space form has positive measure. In case of an immersed sphere our result can be seen as a generalization of the celebrated Hopf theorem.



中文翻译:

在三维空间形式的浸没表面的脐带集合上

我们证明,在关于平均曲率和无痕第二基本形式的某些积分不等式下,浸没表面在三维空间形式中的脐点集具有正度量。在沉浸球的情况下,我们的结果可以看作是著名的霍普夫定理的推广。

更新日期:2020-10-30
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